# Questions tagged [normed-spaces]

A vector space $E$, generally over the field $\mathbb R$ or $\mathbb C$ with a map $\lVert \cdot\rVert\colon E\to \mathbb R_+$ satisfying some conditions.

9,374 questions
Filter by
Sorted by
Tagged with
20 views

34 views

### Practical uses of $p$ norms for $p\notin \{1,2,\infty\}$?

We all love normed spaces, but it seems like the $1$-, $2$-, and $\infty$-norms get the lion's share of the love. That's not admittedly not without good reason, but what of the other unsung norms with ...
70 views

### The space of Lipschitz continuous functions is dense in that of uniformly continuous functions?

Let $(X,d)$ be a metric space. Then The space of bounded uniformly continuous functions is dense in that of bounded continuous functions w.r.t. the supremum norm. ref The space of bounded Lipschitz ...
41 views

### For $p\ge 1$, $x\ge 0$ and $y\ge 0$, $|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$?

I vaguely remember seeing an inequality of the form: $$|x^{\frac{1}{p}}-y^{\frac{1}{p}}|^p \le |x-y|$$ for $p\ge 1$, $x\ge 0$ and $y\ge 0$. Is this correct? If so, how is it proven? Can it be ...
32 views

### Derivative of l2-norm w.r.t matrix

I have a matrix A which is size of mm, a vector b which is size of m1. I'd like to derivative of the following function f(A) w.r.t A: $$f(A)=||A*b||_2$$ I need to find the first order Taylor expansion ...
764 views

24 views

### Derivative of Frobenius norm with respect to scalar

Can anyone explain to me why $$\frac{d}{d\theta }\left\|\textbf{Aw}-\theta(1-c)\textbf{1} \right\|^{2}= 0$$ is equivalent to $$2(1-c)\textbf{1}^T(\theta(1-c)\textbf{1}-\textbf{Aw})=0$$? I'm not very ...
25 views

36 views

### How to prove that normed space is complete?

$I=[0,1]$. For $k \in \mathbb{N}$, denote by $C^k(I)$ the space of real-valued functions on $I$ possessing continuous derivatives up to order $k$ on $I$, including one-sided derivative at the end ...
5 views

### minimal induced norm on sobolev space

I am new to functional analysis and induced norms. I have an exercise question in my university course, and I am trying a lot to get a solution for it. It is a challenging question for me. Could ...
1 vote
21 views

### Functional analysis - best approximation norm [closed]

I have no idea how to prove or disprove the parts of the below question. I intuitively feel they are correct but cannot form a rigorous mathematical proof for it. Given is a normed space E. Also, ...
71 views

### ways to calculate norm of the Bounded Linear Functional.

$\Lambda$ be a linear functional on $C([0,1])$ defined by $$\Lambda(f) = \int_0^1 xf(x)dx \;\;\; \text{ for } f \in C([0,1]).$$ and use $\|f\|_{sup} = \sup_{x \in [0,1]} |f(x)|$ for $f \in C([0,1])$...
1 vote
35 views

### On the Interpretation of Volterra's theory of Integral Equations

Consider an integral equation $$u(x) - \int_a^x K(x,y) u(y) dy = f(x), \qquad x \in [a,b] \qquad (1)$$ where $u:[a,b] \to \mathbb{R}$ is an unknown function and $f$ and $K$ are known continuous ...
24 views

### Complexity of Lebesgue measurable spaces

Consider a discrete finite set $\Omega=X\times Y \in \mathbb{R}^{m\times n}$ for finite $m,n$. Let $(\Omega,\Sigma,\mu)$ be the measure space. ($\Sigma$ is the power set and $\mu$ is $\sigma$-finite ...